Strong and powerful P-tableaux are conjectured to give lower and upper bounds for e-coefficients of chromatic symmetric functions, with exact interpretations proven for several families.
A geometric realization of the chromatic symmetric function of a unit interval graph
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Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. ${\bf 295}$ (2016), ${\bf 329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety $\mathscr X_\Psi$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.
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Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis
Strong and powerful P-tableaux are conjectured to give lower and upper bounds for e-coefficients of chromatic symmetric functions, with exact interpretations proven for several families.